Question

Write the addition and multiplication tables for \(\mathbb{Z}_{3}\).

Short answer

The addition table for \(\mathbb{Z}_{3}\) is:\n\n\[, +) |0\, |1\, |_2\, |\n\|0\, |0\, |1\, | 2\, |\n\|1\, |1\, | 2\, | 0\, |\n\|2\, | 2\, |0\, | 1\, |\]\n And the multiplication table for \(\mathbb{Z}_{3}\) is:\n\n\[, *) | 0\, | 1\, | 2\, |\n\| 0 | 0\, | 0\, | 0\, |\n\| 1\, | 0\, | 1\, | 2\, |\n\| 2\, | 0\, | 2\, | 1\, |\]

Step-by-step solution

Addition table

List all the elements of \(\mathbb{Z}_{3}\): {0, 1, 2}, then perform addition operation on each pair and apply modulo 3 operation to the result. The addition table for \(\mathbb{Z}_{3}\) would look like this:\n\n\[, +)\, |0\, |1\,| 2\,|\n\|0\,\,\, |0\, |1\, |2\, |\n\|1\,\,\, |1\, |2\, |0\, |\n\|2\,\,\, |2\, |0\, |1\, |\]

Multiplication table

The process is similar for the multiplication table. Perform multiplication operation on each pair and apply modulo 3 operation to the result. The multiplication table for \(\mathbb{Z}_{3}\) would look like this:\n\n\[, * )\, | 0\, | 1\, | 2\, |\n\| 0\,\,\, | 0\, | 0\, | 0\, |\n\| 1\,\,\, | 0\, | 1\, | 2\, |\n\| 2\,\,\, | 0\, |2\, |1\, |\]